The aim of this paper is the classi"cation of two-weight irreducible cyclic codes. Using Fourier transforms and Gauss sums, we obtain necessary and su$cient numerical conditions for an irreducible cyclic code to have at most two weights. This gives a uni"ed explanation for all two-weight irreducible cyclic codes and allows a conjecturally complete classi"cation. Aside from the two known in"nite families of two-weight irreducible cyclic codes, a computer search reveals 11 sporadic examples. We conjecture that these are already all two-weight irreducible cyclic codes and give a partial proof of our conjecture conditionally on GRH.2002 Elsevier Science
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